Question

The diameters of bearings used in an aircraft landing gear assembly have a standard deviation of ???? = 0.0020 cm. A random sample of 15 bearings has an average diameter of 8.2535 cm. Please (a) test the hypothesis that the mean diameter is 8.2500 cm using a two-sided alternative and ???? = 0.05; (b) find P-value for the test; and (c) construct a 95% two-sided confidence interval on the mean diameter.

Answer 1

Answer:

(a) We conclude after testing that mean diameter is 8.2500 cm.

(b) P-value of test = 2 x 0.0005% = 1 x $10^{-5}$ .

(c) 95% confidence interval on the mean diameter = [8.2525 , 8.2545]

Step-by-step explanation:

We are given with the population standard deviation, $\sigma$ = 0.0020 cm

Sample Mean, Xbar = 8.2535 cm   and Sample size, n = 15

(a) Let Null Hypothesis, $H_0$ : Mean Diameter, $\mu$ = 8.2500 cm

 Alternate Hypothesis, $H_1$ : Mean Diameter,$\mu$ $\neq$ 8.2500 cm{Given two sided}

So, Test Statistics for testing this hypothesis is given by;

                           $\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }$ follows Standard Normal distribution

After putting each value, Test Statistics = $\frac{8.2535-8.2500}{\frac{0.0020}{\sqrt{15} } }$ = 6.778

Now we are given with the level of significance of 5% and at this level of significance our z score has a value of 1.96 as it is two sided alternative.

Since our test statistics does not lie in the rejection region{value smaller than 1.96} as 6.778>1.96 so we have sufficient evidence to accept null hypothesis and conclude that the mean diameter is 8.2500 cm.

(b) P-value is the exact % where test statistics lie.

For calculating P-value , our test statistics has a value of 6.778

So, P(Z > 6.778) = Since in the Z table the highest value for test statistics is given as 4.4172  and our test statistics has value higher than this so we conclude that P - value is smaller than 2 x 0.0005% { Here 2 is multiplied with the % value of 4.4172 because of two sided alternative hypothesis}

Hence P-value of test = 2 x 0.0005% = 1 x $10^{-5}$ .

(c) For constructing Two-sided confidence interval we know that:

    Probability(-1.96 < N(0,1) < 1.96) = 0.95 { This indicates that at 5% level of significance our Z score will lie between area of -1.96 to 1.96.

P(-1.96 <  $\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }$ < 1.96) = 0.95

P($-1.96\frac{\sigma}{\sqrt{n} }$ < $Xbar - \mu$ < $1.96\frac{\sigma}{\sqrt{n} }$ ) = 0.95

P($-Xbar-1.96\frac{\sigma}{\sqrt{n} }$ < $-\mu$ < $1.96\frac{\sigma}{\sqrt{n} }-Xbar$ ) = 0.95

P($Xbar-1.96\frac{\sigma}{\sqrt{n} }$ < $\mu$ < $Xbar+1.96\frac{\sigma}{\sqrt{n} }$) = 0.95

So, 95% confidence interval for $\mu$ = [$Xbar-1.96\frac{\sigma}{\sqrt{n} }$ , $Xbar+1.96\frac{\sigma}{\sqrt{n} }$]

                                                        = [$8.2535-1.96\frac{0.0020}{\sqrt{15} }$ , $8.2535+1.96\frac{0.0020}{\sqrt{15} }$]

                                                        = [8.2525 , 8.2545]

Here $\mu$ = mean diameter.

Therefore, 95% two-sided confidence interval on the mean diameter

           =  [8.2525 , 8.2545] .

Answer 2

Answer:

a

b

c

Step-by-step explanation: